Local Automorphisms and Derivations in Algebraic Structures

Local automorphisms and derivations capture how global symmetries and infinitesimal transformations of an algebra can be recovered from their action on individual elements. Whereas an automorphism or derivation is a map defined once and for all on the entire algebra, its local counterpart need only agree with some genuine automorphism or derivation at each single point. Recent work has shown that, in a wide variety of settings—from associative matrix algebras and operator algebras to Lie and Lie superalgebras—every local or 2-local map is in fact global. This rigidity lends powerful structural insight and underpins applications ranging from classification theory to quantum information. Concrete instances include generalised matrix algebras, von Neumann algebras, and model filiform Lie superalgebras, where hypotheses on idempotents, topology or grading ensure that any map which “locally” preserves products or brackets must do so uniformly. The field thus bridges classical ring theory, functional analysis and graded algebra, highlighting deep connections between local behaviour and global algebraic shape.

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Technical terms

Local automorphism: A linear map on an algebra that, for each element, agrees with some genuine automorphism, although not necessarily a single one throughout.

Local derivation: A linear map that for each element coincides with some derivation, without a requirement of global consistency.

Lie algebra: An algebra with a bilinear, antisymmetric bracket satisfying the Jacobi identity, modelling infinitesimal symmetries.

Lie superalgebra: A Z₂-graded extension of a Lie algebra with even and odd parts, where the bracket respects grading and satisfies graded Jacobi identities.

Generalised matrix algebra: An algebra built from block matrices over possibly distinct rings or algebras, serving as a unifying framework for many classical structures.

References

1. Local Lie derivations of generalized matrix algebras. AIMS Mathematics (2023).
2. Local Automorphisms and Local Superderivations of Model Filiform Lie Superalgebras. Journal of Mathematics (2024).
3. Characterizations of local Lie derivations on von Neumann algebras. AIMS Mathematics (2022).